UNIT I VECTOR CALCULUS

1.1 Gradient, divergence and curl - Directional derivative - Irrotational and solenoidal vector fields

1.2 Vector integration

1.3 Green’s theorem in a plane

1.4 Gauss divergence theorem

1.5 Stokes’ theorem

1.6 Simple applications involving cubes and rectangular parallelopipeds

UNIT II ORDINARY DIFFERENTIAL EQUATIONS

2.1 Higher order linear differential equations with constant coefficients

2.2 Method of variation of parameters

2.3 Cauchy’s and Legendre’s linear equations

2.4 Simultaneous first order linear equations with constant coefficients.

UNIT III LAPLACE TRANSFORM

3.1 Laplace transform

3.2 Sufficient condition for existence and Transform of elementary functions

3.3 Basic properties

3.4 Transforms of derivatives and integrals of functions

3.5 Transforms of unit step function and impulse functions

3.6 Transform of periodic functions

3.7 Inverse Laplace transform

3.8 Statement of Convolution theorem

3.9 Initial and final value theorems

3.10 Solution of linear ODE of second order with constant coefficients using Laplace transformation techniques

UNIT IV ANALYTIC FUNCTIONS

4.1 Functions of a complex variable

4.2 Analytic functions: Necessary conditions

4.3 Cauchy-Riemann equations and sufficient conditions

4.4 Harmonic and orthogonal properties of analytic function

4.5 Harmonic conjugate

4.6 Construction of analytic functions

4.7 Conformal mapping: w =z+k, kz, 1/z, z2, ez , Bi-linear transformation

UNIT V COMPLEX INTEGRATION

5.1 Complex integration

5.2 Statement and applications of Cauchy’s integral theorem and Cauchy’s integral formula

5.3 Taylor’s and Laurent’s series expansions

5.4 Singular points

5.5 Residues

5.6 Cauchy’s residue theorem

5.7 Evaluation of real definite integrals as contour integrals around unit circle and semi-circle